296. Global aspects of the scalar meson puzzle
HEP-TH/PH — By Renata Jora on March 7, 2009 at 4:05 pm
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This is a guest post by Renata Jora from INFN. Dmitry.
I would like to thank Dmitry for inviting me to write this post about our paper (A. Fariborz, R. Jora and J. Schechter) “Global aspects of the scalar meson puzzle”, arXiv:0902.2825.
I will start with a few historical facts. It was first noticed that the strong interaction is approximately blind at the interchange of the neutron to the proton which lead to the introduction of an 
invariance. Then another approximate symmetry was discovered, the hypercharge 
where 
is the baryon number and 
is the strangeness. As one tries to incorporate both of this groups into a larger one obtains the invariance under an 
. Note that all these groups act in the flavor space. This leads to the the “constituent quark model” (Gell Mann) which states that quarks have different quantum numbers and the meson are consequently made out of one quark and one antiquark, while the baryons have a three quark structure. All these combinations should be color neutral. If all quark masses are set to zero the actual invariance group is a larger one, the chiral 
group. However since this symmetry is not realized in the spectrum it is then assumed that is spontaneously broken. According to the Goldstone theorem this would lead to 8 massless mesons. In reality there are very light particles like the pions but none of them is massless which means that this chiral symmetry should be also explicitly broken by the quark masses.
If one further analyzes the mass spectrum of the pseudoscalar (
), scalar (
) and vector mesons, one finds that for example for the vector meson nonet:
the masses increase with the strange quark content:
The same mass ordering is approximately valid for the pseudoscalar mesons. However if one assumes for the light scalar mesons a diquark structure one gets:
It is evident that the previous mass ordering is not respected. This is the core of the scalar meson puzzle. It was observed long time ago that this problem might be solved by assuming that the light scalar mesons have some four quark content. Moreover there are more scalar states that one can fit into a multiplet (one can actually form two multiplets). Inspired by this, in the paper “Global aspects of the scalar meson puzzle” (arXiv:0902.2825) we consider a linear sigma model with global invariance and with two nonets (
and 
where , 
are the scalars and 
, 
are the pseudoscalars) one with a diquark structure the other one with a four quark structure. The symmetry is broken explicitly by an SU(2) invariant term. The corresponding Lagrangian has the form
,
where 
stands for a function made from 
(but not necessarily 
) invariants formed out of 
and 
.
Here
.
We chose the terms such that the number of quark lines at each vertex 
does not exceed 8. All the terms except the last one also possess the invariance. The symmetry breaking term which models the QCD mass term takes the form:
where 
are proportional to the three light quark masses.
The model allows for two-quark condensates, 
as well as four-quark condensates 
. Here we assume isotopic spin symmetry so 
and
We also need the “minimum” conditions,
There are 12 parameters in the model describing the Lagrangian and the vacuum. Together with the four minimum conditions this reduces the number of necessary inputs to 8. These are:
Because 
has such a large uncertainty, we examine predictions depending on the choice of this mass within its experimental range. The sixth input will be taken as the light “quark mass ratio” 
, which will be varied over an appropriate range.
Thus we are left with the problem of determining 9 masses and 16 four quark percentages. The main result of the paper is that while the lightest pseudoscalars have low four quark percentage thus being mainly quark-antiquark structures as expected the low lying scalars have large four quark percentage (larger than 
percent).
How can this resolve the scalar meson puzzle? Let us take a closer look at the I=1 scalars for example, the a’s. In the quark antiquark picture as we previously mentioned the a’s were composed of u and d quarks fact which could not explain their relative heaviness. However in our model the a’s correspond to an admixture of 
and 
. Let us assume the molecule picture for the four quark states, i.e.
.
Then simply 
and consequently will contain 
which is associated with 
. Thus a large four quark percentage in this case correspond with large content.
9 Comments
Dear Renata,
I would like to point out to you the work of Narison, Ochs and others claiming that the gamma-gamma width for the sigma is to small to be a tetraquark state (e.g. see http://arxiv.org/abs/0804.4452 ). I think that nuclear forces can grant a tetraquark state just for the heaviest quarks being in this case almost Coulombian and granting a residual molecular force.
Marco
Dear Renata,
Are you also saying that pions consist of four quarks (or at least have four quark content)? If yes, it differs quite a bit from what Jaffe wrote in his 1977 paper you cite (he expected that only
, 
, 
, 
have 4-quark content among light mesons).
Cheers,
Dmitry.
Dear Marco,
Thank you for pointing out this paper to me. Cheers,
Renata
Dear Dmitry,
In our model(linear sigma model with two chiral nonets) unlike Jaffe which takes into account only some light mesons or ‘t Hooft, Isidori, Maiani, Polosa, Riquer,Phys Lett B662,424 which consider that the pseudoscalars have a two quark content and only the scalars have a four quark structure, we make the assumption that both the pseudoscalars(pions) and scalars are an admixture of two quark and four quark components. Thus the pions have a four quark component, however small.
Cheers,
Renata
Dear Renata,
thanks! I thought it is a bit unnatural to expact that lighter mesons (note that pion mass is
and goes to zero at 
) consist of larger number of quarks…
By the way, what is the physical origin of the
term in the potential? I think I understand where all other terms might come from but not this one.
Cheers,
Dmitry.
Dear Dmitry,
The c_3 term is chosen to model the U(1)_A violating part in the effective lagrangian. See for example arXiv:0801.2552 . ‘t Hooft first determined such a quark term for the case of two flavors.(Phys Rev D 14, 3432(1976)).
Cheers,
Renata
Dear Renata,
thanks for the reference! Suppose you restore spontaneously broken chiral symmetry (set
to zero). What happens in the model you describe? Presumably, masses of all states should go to zero in the approximation you use (QCD non-pert. physics is neglected, so 
cannot appear in your formulae).
Cheers,
Dmitry.
Dear Dmitry,
As one puts to zero the explicit symmetry breaking term one still can have spontaneous symmetry breaking. See for example the classic SU(2)_L\times SU(2)_R linear sigma model. There the spontaneous symmetry breaking leads to massless pions and a massive sigma.
Our model uses mostly the symmetries of the low energy QCD. It so happens that symmetries are a very important part of physics.
Cheers,
Renata
Oh, right, that was explicit symmetry breaking term… Somehow, I misunderstood and thought that it is related to spontaneous chiral symmetry breaking. Thanks!
Dmitry.
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